Stability of the Augmented Solow Model with Physical and Human Capital Diffusion
Keywords:
augmented Solow model, reaction–diffusion, stability, bifurcation, balanced growth pathAbstract
This study formulates the augmented Solow model as a two-component reaction–diffusion system, governing the spatial dynamics of physical and human capital per effective unit of labor. The analysis examines the local asymptotic stability of the positive steady state, the effect of spatial capital diffusion on stability properties, and the possibility of Turing and Hopf bifurcations under decreasing returns to scale. Stability is analyzed using linearization and Jacobian-based stability criteria, including the Routh–Hurwitz conditions, for both the non-spatial and spatial systems. The spatial Jacobian is shown to satisfy a strictly negative trace and a strictly positive determinant for all nonnegative wave numbers, confirming that the steady state remains locally asymptotically stable. Bifurcation analysis further establishes that neither Turing nor Hopf instabilities arise, demonstrating that spatial diffusion does not destabilize the balanced growth path. Numerical simulations using a forward Euler finite-difference scheme illustrate convergence of both capital variables to the steady state under simultaneous localized perturbations. These results indicate that, for initial conditions within a neighborhood of the steady state, the joint mobility of physical and human capital supports convergence toward the homogeneous balanced growth path, with each spatial mode decaying toward the steady state, while decreasing returns to scale prevent persistent spatial disparities and oscillatory behavior.
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